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WHAT THEY USE TO SEE WHAT THEY SEE AND DO WHAT THEY DO

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. TELESCOPES .The basic telescope used by the students at Hogwarts and amateur muggle astronomers are likely very similar; wizarding telescopes seem to be closer to the muggle turn-of-the-century units that their modern counter parts however - at least in appearance.
How Powerful Are They?Just from the book's text, it is hard to judge how powerful the 'average' wizarding telescope is, as Rowling's descriptions of what the children are seeing is vague at best (and sometimes incorrect at worst, but that's a different topic altogether). It is likely that they are at the very least as powerful as the the average muggle amateur telescope, since in the books they can apparently see things in more detail 'off the shelf' telescope on Earth can see. However, it is unclear in the books if these views come from their (the student's) telescopes or from reference books used by students, which could actually being using
muggle images, as the images in question (those of Io, one of the Galilean moons of Jupiter) were taken in 1979. These could also be from wizarding Astronomers. Unfortunately, there's no way to tell.
Professional wizarding astronomers likely have better telescopes than those of students, though again, how good is basically unknown. Wizards seem to show little interest in the universe beyond the solar system, from what we are shown in the books - whether this means they have yet to adapt the technology to see much, or that their goals (astrology, divination, etc) don't bring them to study the wider universe, is up for debate. So far, out of character, we have yet to reach a consensus. Play as you feel comfortable, at this point - but keep in mind the
Magic and Science document.
What do the Students Use Their Telescopes For?Due to the complexity it ads, students are not expected to use their telescopes in star charting until late into their 3rd year. It becomes a regular tool by the 5th, but would still not be use for the bulk of their basic charting. It comes into play while charting distant planets, for the most part.
What they do use it for, for all 7 years, is study of non-naked eye objects - such as the Galilean moons, Uranus, Neptune, Pluto, many of the Asteroids. Under Trishna they also study Nebulas, Star Clusters and other Galaxies.

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. ASTROLABES .An Astrolabe is an archaic astronomy tool, largely abandoned by muggles in the 16th century. Wizards, however, keep it, as it aids in their goals more so than it does for modern muggles'.
An Astrolabe is, in short...an astronomy, navigation, astrology, surveying, triangulation and divination tool. Its primary function is to aid in predicting the positions of the planets, Sun, Moon, and stars. Note that the usual astrolabe, somewhat unintuatively, is actually rather unusable at sea. A Sextant works better for that purpose.
The Astrolabe is not very useful in modern muggle science, as its common use and creation date to the pre-heliocentric systems. For wizards, they are likely used in ancient forms of divination and astrology. They may also be used in keeping time via the stars, as the Wizarding idea of an astronomical clock is likely structured differently.
They likely see much more use amoung wizards hailing from the Islamic world, where the tool was very commonly used not only for all of the reasons listed above, but to schedule prayers properly.

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. ORRERYS .Orrerys are also archaic, but still serve some use, and are likely quite popular among wizarding children. An Orrery is essentially a small planetarium, showing the orbits of the planets in a Heliocentric model. A clockwork device, they can run on their own even in the muggle world. They were Invented by muggles in 1704.
Like most Astronomy tools listed here, their largest practical use is in divination and astrology, though the Orrery itself is a good educational tool for young children, and that may be its main purpose for the wizard world. While the spacial relationships are not always correct, the properly made modern orrery shows the orbital paths fairly accurately. They can be built quite large, varying from a table top to
an entire ceiling. In modern times, they've been largely replaced by the planetarium.
====================================. SEXTANT (ASTRONOMICAL) .This is a very large tool - a variation of the navigator's sextant, developed specifically with astronomy in mind. Instead of using reflecting mirrors, as the nautical sextant does, these large tools use direct angles to measure the positions and space between astronomical bodies with surprising accuracy.
These tools are extremely large - up to 7 feet in length on each side of the wedge. They often require more than one person to operate unless magic is involved - as minute adjustments have to be made to two separate pieces of the instrument, however, a second person is likely preferable to a spell. Made of brass, wood and often iron, they are hardly portable. Used largely to plot the movement of planets and stars - as well as anything of interest.
====================================. ARMILLARY SPHERES .Armillary Spheres are likely one of the most commonly used tools of wizards. The Armillary sphere is a celetstial globe, and an acient tool of Astronomers worldwide. Their inventor was like Eratosthenes, who lived around 150BCE.
Each of the bands in the Armillary sphere represents and a line in the heavens - the Equatorm the Ecliptic, Meridians, and the Parallels. Old spheres have the Earth at their center, and more modern ones have the sun.
An Armillary sphere aids and Astronomer in findig certain locations and time in the heavens. They are extremely complex machines and not incredibly intuative, as this description of their use can show.
To rectify the sphere for use, first slacken the screw r in the upright stem R, and taking hold of the arm Q, move it up or down until the given degree of latitude for any place be at the side of the stem R; and then the axis of the sphere will be properly elevated, so as to stand parallel to the axis of the world, if the machine be set north and south by a small compass: this done, count the latitude from the north pole, upon the celestial meridian L, down towards the north notch of the horizon, and set the horizon to that latitude; then, turn the nut b until the sun Y comes to the given day of the year in the ecliptic, and the sun will be at its proper place for that day: find the place of the moon's ascending node, and also the place of the moon, by an Ephemeris, and set them right accordingly: lastly, turn the winch W, until either the sun comes to the meridian L, or until the meridian comes to the sun (according as you want the sphere or earth to move) and set the hour-index to the XII, marked noon, and the whole machine will be rectified. — Then turn the winch, and observe when the sun or moon rise and set in the horizon, and the hour-index will show the times thereof for the given day.
The Astronomy classroom was once graced with an extremely large and magically driven armillary sphere, suspended from the cieling of the tower. It turned on its own, seemingly, and followed the current time. Since Professor Trishna arrived, this large sphere has mysteriously vanished, replaced by his own smaller unit. His on sphere is now covered in white fabric, and is actually the globe he uses in classes as a projector of sorts.
Anyone spying the house elves stuffing large brass discs into storage was, of course, mistaken. As was anyone in the Divination rooms who heard a large metal 'bong' and pained cursing several times before the sphere's disappearance.